First I ask myself if what they're doing is really that big of a deal and determine why it is. Usually it isn't as big of a deal as I thought it was and I let them explore as long as I'm there watching.
If I've determined that it's a big deal then I try to phrase things in a positive or informational way while giving a visual cue whenever possible instead of just "no." It worked well with my older daughter (2.5 now) and I'm just starting it with my almost 11 month old daughter.
For example, with hot things, I'm right there if my youngest is near something hot and I tell her that it's hot and would hurt if she touched it. Then I give a visual example by putting my hand close enough to feel the heat radiating off and then pulling it away before touching it. My older daughter mimicked me and only once did she actually try touching something hot (at around 2 years old) and she hasn't done it since. She loves to "help" with the cooking now :) My younger daughter still just watches me do it.
For me, not saying "no" very often has less to do with what I want my girls to say than with the fact that I want them to question most things. I don't want them getting in the habit of just blindly following orders, even from me. My older knows that there are times when I mean business and when she doesn't understand my reasons and I do say "no" if there's danger involved. However, I think that it dilutes the sense of danger if "no" is used for non-dangerous situations as well.
I fully expect their response to a "no" in the future, as they get older and more discerning, to be a "why" which is great, in my opinion!
I also tell them to "hold on tight" or "be careful" instead of "don't fall" because I don't even want to put the idea of falling into their heads. So I'm using the positive form (what TO do as opposed to what NOT to do) whenever possible.
Best wishes!
~B.
ETA - I also used the phrase, "that's not yours! That's mama's/papa's" quite a lot while removing them or the item if they were getting into my things or my husband's things. I want to instill in my children a sense of respect for others and others' belongings. That phrase also worked much better than a "no" since it involved giving a reason and then physically removing them or the item gave concrete reinforcement to the abstract statement.